Question

Two temperature measurements are taken with a thermometer marked with the Celsius scale. Show that the difference between the two readings would be the same if the temperatures were converted to the Kelvin scale.

Short answer

The temperature difference is the same in both Celsius and Kelvin scales.

Step-by-step solution

Understand the relation between Celsius and Kelvin scales

Recall that temperature in Kelvin (\text{K}) is related to temperature in Celsius (\text{C}) by the formula: \[ T_{\text{K}} = T_{\text{C}} + 273.15 \]

Define the two temperature measurements

Let the two temperature measurements in Celsius be \[ T_{\text{C1}} \] and \[ T_{\text{C2}} \]

Convert the Celsius measurements to Kelvin

Using the relation from Step 1, convert \[ T_{\text{C1}} \] and \[ T_{\text{C2}} \] to Kelvin: \[ T_{\text{K1}} = T_{\text{C1}} + 273.15 \] \[ T_{\text{K2}} = T_{\text{C2}} + 273.15 \]

Calculate the difference on the Celsius scale

Find the difference between the two temperatures in Celsius: \[ \text{Difference in } ^\text{C} = T_{\text{C2}} - T_{\text{C1}} \]

Calculate the difference on the Kelvin scale

Similarly, find the difference between the two temperatures in Kelvin: \[ \text{Difference in K} = T_{\text{K2}} - T_{\text{K1}} \]

Show the differences are equal

Substitute the converted temperatures: \[ \text{Difference in K} = (T_{\text{C2}} + 273.15) - (T_{\text{C1}} + 273.15) \] Simplify the equation: \[ \text{Difference in K} = T_{\text{C2}} - T_{\text{C1}} \] Therefore, the difference in Kelvin is equal to the difference in Celsius.

Key concepts

Celsius Scale

The Celsius scale is a temperature measurement scale commonly used around the world. On this scale, water freezes at 0°C and boils at 100°C under standard atmospheric pressure. The Celsius scale is designed to be intuitive, since it is based on the everyday experience of water. It is often used in most scientific and metric-aligned countries. To calculate temperature differences in Celsius, simply subtract one Celsius value from another, which is straightforward and easy to comprehend.

Kelvin Scale

The Kelvin scale is another important temperature scale used mainly in scientific contexts. It starts from absolute zero, the point at which all molecular motion ceases, which is 0 K. Unlike the Celsius scale, there's no degree symbol with Kelvin. The Kelvin scale is related to the Celsius scale by the relation: \( T_{K} = T_{C} + 273.15 \). This means that zero degrees Celsius is equivalent to 273.15 K. The Kelvin scale is used in various scientific calculations because it provides an absolute reference point for temperature measurements.

Temperature Difference

Understanding temperature difference is crucial in various scientific applications. The difference between two temperatures is calculated the same way whether you use the Celsius or Kelvin scales. This is because both scales are linearly related by a constant 273.15. For example, if you have two temperatures \( T_{C1} = 25 \)°C and \( T_{C2} = 35 \)°C, converting them to Kelvin would result in \( T_{K1} = 298.15 \) K and \( T_{K2} = 308.15 \) K respectively. The difference in Celsius is \( 35 - 25 = 10 \)°C, and likewise, the difference in Kelvin is \( 308.15 - 298.15 = 10 \) K. Thus, temperature differences remain the same when converting between Celsius and Kelvin, demonstrating that the scales are consistent in measuring change in temperature.